Handy Polynomial Fitting with Bernstein Polynomials

[Thomas Klemm]

(11-11-2018 03:10 AM)Valentin Albillo Wrote:  What values does produce your Bernstein fit for, say, order 2 ?
What are the coefficients of the Bernstein Polynomial fit ?

The Bernstein Polynomial reproduces the function for \(f(x)=1\) and \(f(x)=x\):

\(
\begin{matrix}
B_n(1;x)=1 \\
B_n(x;x)=x
\end{matrix}
\)

However for \(f(x)=x^2\) we get:

\(
B_n(x^2;x) = x^2 + \frac{1}{n}x(1−x)
\)

Thus for \(n=10\) it's off by \(\frac{1}{40}\) at \(x=\frac{1}{2}\).

While it can be shown that \(\lim _{n\to \infty }{B_{n}(f)}=f\) uniformly on the interval [0, 1] the convergence is rather slow.

Cheers
Thomas